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12th Standard Maths English Medium Theory of Equations Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  1. A zero of x3 + 64 is

    (a)

    0

    (b)

    4

    (c)

    4i

    (d)

    -4

  2. If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

    (a)

    mn

    (b)

    m+n

    (c)

    mn

    (d)

    nm

  3. The number of real numbers in [0,2π] satisfying sin4x-2sin2x+1 is

    (a)

    2

    (b)

    4

    (c)

    1

    (d)

  4. If x3+12x2+10ax+1999 definitely has a positive zero, if and only if

    (a)

    a≥0

    (b)

    a>0

    (c)

    a<0

    (d)

    a≤0

  5. The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)rxr is

    (a)

    0

    (b)

    n

    (c)

    < n

    (d)

    r

  6. If a, b, c ∈ Q and p +√q (p,q ∈ Q) is an irrational root of ax2+bx+c=0 then the other root is

    (a)

    -p+√q

    (b)

    p-iq

    (c)

    p-√q

    (d)

    -p-√q

  7. Let a > 0, b > 0, c >0. h n both th root of th quatlon ax2+b+C= 0 are

    (a)

    real and negative

    (b)

    real and positive

    (c)

    rational numb rs

    (d)

    none

  8. lf the root of the equation x3 +bx2+cx-1=0 form an lncreasing G.P, then

    (a)

    one of the roots is 2

    (b)

    one of the rots is 1

    (c)

    one of the rots is -1

    (d)

    one of the rots is -2

  9. For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has

    (a)

    one solution

    (b)

    two solution

    (c)

    at least two solution

    (d)

    no solution

  10. If the equation ax2+ bx+c=0(a>0) has two roots ∝ and β such that ∝<- 2 and β > 2, then

    (a)

    b2-4ac=0

    (b)

    b2 - 4ac <0

    (c)

    b2 - 4ac >0

    (d)

    b2 - 4ac≥0

  11. If (2+√3)x2-2x+1+(2-√3)x2-2x-1=\(\frac { 2 }{ 2-\sqrt { 3 } } \) then x=

    (a)

    0,2

    (b)

    0,1

    (c)

    0,3

    (d)

    0, √3

  12. If ∝, β,૪ are the roots of 9x3-7x+6=0, then ∝ β ૪ is __________

    (a)

    \(\frac{-7}{9}\)

    (b)

    \(\frac{7}{9}\)

    (c)

    0

    (d)

    \(\frac{-2}{3}\)

  13. If x2 - hx - 21 = 0 and x2 - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

    (a)

    0

    (b)

    1

    (c)

    4

    (d)

    3

  14. If ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

    (a)

    c>0

    (b)

    c<0

    (c)

    c=0

    (d)

    c≥0

  15. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

    (a)

    no

    (b)

    1

    (c)

    2

    (d)

    infinite

    1. 2 Marks


    10 x 2 = 20
  16. If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

  17. Show that the equation 2x2−6x+7=0 cannot be satisfied by any real values of x.

  18. If x2+2(k+2)x+9k=0 has equal roots, find k.

  19. Solve: (2x-1)(x+3)(x-2)(2x+3)+20=0

  20. Determine the number of positive and negative roots of the equation x9-5x4-14x7=0.

  21. Construct a cubic equation with roots 1,1, and −2

  22. Construct a cubic equation with roots 2,−2, and 4.

  23. Discuss the nature of the roots of the following polynomials:
    x5-19x4+2x3+5x2+11

  24. Find th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

  25. Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)

    1. 3 Marks


    10 x 3 = 30
  26. If α and β are the roots of the quadratic equation 2x2−7x+13 = 0 , construct a quadratic equation whose roots are α2 and β2.

  27. If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

  28. If α, β and γ are the roots of the cubic equation x3+2x2+3x+4=0, form a cubic equation whose roots are, 2α, 2β, 2γ

  29. Find the sum of squares of roots of the equation 2x4-8x+6x2-3=0.

  30. Find the condition that the roots of x3+ax2+bx+c = 0 are in the ratio p:q:r.

  31. If p is real, discuss the nature of the roots of the equation 4x2+4px+p+2=0 in terms of p.

  32. If α, β, and γ are the roots of the polynomial equation ax3+bx2+cx+d=0 , find the value of \(\Sigma \frac { \alpha }{ \beta \gamma } \) in terms of the coefficients.

  33. Solve the cubic equations:
    2x3-9x2+10x=3

  34. Solve the following equations,
    12x+8x=29x2-4

  35. Solve the equations
    x4+3x3-3x-1=0

    1. 5 Marks


    7 x 5 = 35
  36. Form the equation whose roots are the squares of the roots of the cubic equation x3+ax2+bx+c = 0.

  37. Solve the equation x3−9x2+14x+24=0 if it is given that two of its roots are in the ratio 3:2.

  38. If 2+i and 3-\(\sqrt{2}\) are roots of the equation x6-13x5+62x4-126x3+65x2+127x-140=0, find all roots.

  39. Solve:
    (x-5)(x-7)(x+6)(x+4)=504

  40. If a, b, c, d and p are distinct non-zero real numbers such that (a2+b2+c2) p2-2 (ab+bc+cd) p+(b2+c2+d2)≤ 0 the n. Prove that a,b,c,d are in G.P and ad=bc

  41. If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p. 

  42. Solve: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.

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